The subvariety of commutative residuated lattices represented by twist-products

Abstract

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety KK of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of KK , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in KK , and we analyze the subvariety of representable algebras in KK . Finally, we consider some specific class of bounded integral commutative residuated lattices GG , and for each fixed element L∈GL∈G , we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras.